The delineation of clusters of contiguous objects has been discussed in Section 12.6 for time series and spatial transects. The method of chronological clustering, in particular, was described in Subsection 12.6.4; it proceeds by imposing to a clustering algorithm a constraint of contiguity along the time series. Constraints of contiguity have been applied to spatial clustering by several authors, including Lefkovitch (1978,

1980), Monestiez (1978), Lebart (1978), Roche (1978), Perruchet (1981) and Legendre & Legendre (1984c). In the present Subsection, it is generalized to two- or three-dimensional spatial data and to spatio-temporal data.

Constrained clustering differs from its unconstrained counterpart in the following way.

• Unconstrained clustering methods (Chapter 8) only use the information in the similarity or distance matrix computed among the objects. In hierarchical methods, a local criterion is optimized at each step; in all methods included in the Lance and Williams general model, for instance, the objects or groups clustered at each step are those with the largest fusion similarity or the smallest fusion distance. In partitioning methods, a global criterion is optimized; in K-means, for instance, the algorithm looks for K groups that feature the smallest sum of within-group sums-of-squares EK.

• Constrained clustering methods take into account more information than the unconstrained approaches. In the case of spatial or temporal contiguity, the only admissible clusters are those that obey the contiguity relationship. Spatial contiguity may be described by one of the connecting schemes of Subsection 1. The criterion to be optimized during clustering is relaxed to give priority to the constraint of spatial contiguity. It is no surprise, then, that a constrained solution may be less optimal than its unconstrained counterpart in terms of the clustering criterion, e.g. EK. This is balanced by the fact that the solution is likely to more readily interpretable.

It is fairly easy to modify clustering algorithms to incorporate a constraint of spatial contiguity (Fig. 13.23). As an example, consider the clustering methods included in the Lance and Williams general clustering model (Subsection 8.5.9). At the beginning of the clustering process, the vector of group membership has each object as a different group. Proceed as follows:

1. Compute a similarity matrix among objects, using the non-geographic information.

2. Choose a connecting scheme (Subsection 1) and produce a list of connection edges as in Figs. 13.20 and 13.21. Read in the file of edges and transform it into a contiguity matrix containing 1's for connected sites and 0's elsewhere.

3. Compute the Hadamard product of these two matrices. The Hadamard product of two matrices is the product element by element. The resulting matrix contains similarity values in the cells where the contiguity matrix contained 1's, and 0's elsewhere.

4. The largest similarity value in the matrix resulting from step 3 determines the next pair of objects or groups (h and i) to be clustered. Modify the vector of group membership (right of the Figure), giving the same group label to all members of former groups h and ,.

5. Update the similarity matrix using eq. 8.11.

6. Update also the contiguity matrix. All objects that were neighbours to h are now also neighbours to i and vice versa.

7. Go back to step 3. Iterate until all objects are members of a single group.

Ferligoj & Batagelj (1982) have shown, however, that the introduction of relational constraints (e.g. spatial contiguity) may occasionally produce reversals with any of the hierarchical clustering methods included in the Lance & Williams algorithm (Subsection 8.5.9], except complete linkage. Additional constraints may be added to the algorithm, for example to limit the size or composition of any group (Gordon, 1996c). A"-means partitioning algorithms (Section 8.8) may also be constrained by the contiguity matrix shown in Fig. 13.23.

Spatially constrained clustering is useful in a variety of situations. Here are some examples.

• In many studies, there are compelling reasons to force the clusters to be composed of contiguous sites; for instance, when delineating ecological regions, political voting units, or resource distribution networks.

• One may wish to relate the results of clustering to geographically-located potential causal factors that are known to be spatially autocorrelated, e.g. geological data.

• One may wish to cluster sites based upon physical variables, using a constraint of spatial contiguity, in order to design a stratified biological sampling program to study community composition.

• To test the hypothesis that neighbouring sites are ecologically similar, one may compare unconstrained and constrained clustering solutions using the modified Rand index (Subsection 8.11.2). De Soete et al. (1987) give other examples where such comparisons may help test hypotheses in the fields of molecular evolution, psycholinguistics, cognitive psychology, and evolution of languages.

• Constrained solutions are less variable than unconstrained clustering results, which may differ in major ways among clustering methods. Indeed, the constraint of spatial contiguity reduces the number of possible solutions and forces different clustering algorithms to converge onto largely similar clusters (Legendre et al., 1985).

Constrained clustering may also be used for three-dimensional or spatio-temporal sampling designs (e.g. Planes et al., 1993). As long as the three-dimensional or spatiotemporal contiguity of the observations can be accurately described as a file of edges as in Figs. 13.20 and 13.21, constrained clustering programs have no difficulty in computing the solution; the only difficulty is the representation of the results as three-dimensional or spatio-temporal maps. Higher-dimensional extensions of the geometric connecting schemes presented in Subsection 1 are available if required.

Legendre (1987b) suggested a way of introducing spatial proximity into clustering algorithms which is less stringent than the methods described above. The method consists in weighting the values in the ecological similarity or distance matrix by some function of the geographic distances among points, before clustering. The idea was taken up by Bourgault et al. (1992) who proposed to use a multivariate variogram or covariogram as spatial weighting function prior to clustering. Large ecological distances between sites that are close in space are downweighted to some extent by this procedure. It is then easier for clustering algorithms to incorporate somewhat diverging sites into neighbourhood clusters. Oliver & Webster (1989) suggested to use a univariate variogram for the same purpose.

Constrained classification methods have recently been reviewed by Gordon (1996c). Formal aspects have been discussed by Ferligoj & Batagelj (1982, 1983). Algorithms have been surveyed by Murtagh (1985). Generalized forms of constrained clustering have been described by De Soete et al. (1987).

Numerical example. An artificial set of 16 sites was constructed to represent staggered-row sampling of a distribution with two peaks. From the geographic positions of the sites, a Delaunay triangulation (35 edges) was computed (Fig. 13.24a). A single variable was attributed to the sites. For three groups, the unconstrained K-means solution has a sum of within-group sums-of-squares EK = 53 (Fig. 13.24b). The constrained K-means solution, for three groups,

(a) Delaunay triangulation

(a) Delaunay triangulation

(b) Unconstrained ^-means solution

(c) Constrained ^-means solution

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